An apparatus of the above-mentioned type is known in different embodiments. The filaments are spun first with the help of a spinneret, and then are usually passed down through a cooling chamber. After cooling, the filaments or the filament bundle reach a stretching passage of a stretching unit in which they are stretched aerodynamically. The transverse width of the stretching passage in the machine or travel direction of the spinning fleece is here normally 10-20 mm. Because of this relatively small dimension, the filament bundle in the stretching unit is compacted relatively strongly. This has the disadvantage, in many known installations, that the filaments can be separated only with considerable difficulty for deposition as individual filaments.